Efficiency--Complexity Tradeoff and Constrained Percolation

Overview & Synopsis

We study the efficiency--complexity tradeoff for resource-limited agents through directed first-passage percolation. Nonnegative i.i.d. edge weights represent physical energy expenditure, while the logarithmic size of a constrained path family measures representational complexity. For translation-invariant families, a path-joining property yields deterministic large-distance costs in rational directions and joint convexity in constraint parameter and direction, under suitable moment and feasibility assumptions. Our main result concerns the limiting cost parameterized by path-family entropy. Using techniques from analytic combinatorics in several variables (ACSV), we establish its almost-sure existence and joint continuity in entropy and direction for nested families satisfying additional generating-function conditions, on the corresponding entropy range. Path-count asymptotics supply the change of parameter linking this cost to the constraint-parameterized cost. Two planar examples constrain the number of turns or the number of East-runs of length at least two.